Question

Show that if S is a proposition, where S is the conditional statement “If S is true, then unicorns live,” then “Unicorns live” is true. Show that it follows that S cannot be a proposition. (This paradox is known as Löb’s paradox.)

Short answer

Unicorns live. But we know that unicorns do not live. It follows that S cannot be a proposition.

Step-by-step solution

Introduction

Suppose S: Unicorns live and B: If S is true, then unicorns live.

Thus, B asserts\(B \to S\).

Assumption and proof

Now assume B is false. …… (1)

But S and\(B \to S\)are the same sentence, so that means that the sentence\(B \to S\)is also false. But from equation (1) we get that B is false. That means the assumption in the equation was false.

Conclusion

Hence, B must be true. Equivalently, the sentence \(B \to S\)is true. Now, both of these facts together prove that S is true.

So, we have proved that: Unicorns live. But we know that unicorns do not live. It follows that S cannot be a proposition.